A291698 a(n) = [x^n] Product_{k>=1} (1 + n*x^k).

1, 1, 2, 12, 20, 55, 294, 497, 1224, 2520, 14410, 21912, 54300, 104286, 220710, 1105215, 1697552, 3839382, 7356762, 14873580, 26275620, 132112596, 188666126, 423247104, 772560600, 1535398150, 2632049290, 4975242048, 21273166572, 30649985160, 64824339630, 116604788800, 223181224992

Offset: 0

Author

Ilya Gutkovskiy, Aug 30 2017

Keywords

• nonn
• nice

Comments

The number of partitions of n into distinct parts where each part can be colored in n different ways. For example, there are 4 partitions of 6 into distinct parts, namely 6, 5 + 1, 4 + 2 and 3 + 2 + 1; allowing for the colorings gives a(6) = 6 + 6*6 + 6*6 + 6*6*6 = 294. - Peter Bala, Aug 31 2017

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..2000 from Robert Israel)

Crossrefs

Main diagonal of A286957.

Cf. A022629, A124577, A292190, A292304.

Programs

Maple

seq(coeff(mul(1+n*x^k,k=1..n),x,n),n=0..50); # Robert Israel, Aug 30 2017

Mathematica

Table[SeriesCoefficient[Product[1 + n x^k, {k, 1, n}], {x, 0, n}], {n, 0, 32}]
Table[SeriesCoefficient[QPochhammer[-n, x]/(1 + n), {x, 0, n}], {n, 0, 32}]

Formula

a(n) = A286957(n,n).

a(n) == 0 (mod n); a(n) == n (mod n^2). - Peter Bala, Aug 31 2017

Conjecture: a(n) ~ exp(sqrt(2*(log(n)^2 + Pi^2/3)*n)) * (log(n)^2 + Pi^2/3)^(1/4) / (sqrt(Pi) * (2*n)^(5/4)). - Vaclav Kotesovec, Sep 15 2017